A join theorem for the computably enumerable degrees

Abstract

It is shown that for any computably enumerable (c.e.) degree w \mathbf {w} , if w ≠ 0 \mathbf {w\not =0} , then there is a c.e. degree a \mathbf {a} such that ( a ∨ w ) ′ = a = 0 \mathbf {(a\lor w)’} = \mathbf {a}= \mathbf {0} (so a \mathbf {a} is low 2 _2 and a ∨ w \mathbf {a\lor w} is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low 2 _2 c.e. degrees are not elementarily equivalent as partial orderings.